When Odysseus disguises himself as an old man in disguise why does he need to keep his secret from his wife in The Odyssey?

The section I think you must be refering to is Book 19 of
this excellent epic classic, when Odysseus and Penelope finally are able to meet by
themselves. What is interesting about this meeting is that Odysseus continues to stay in
disguise, tricking his wife into believing he is not who he is and also giving her news
of her husband. Of course, the question we all have is why on earth at this stage in his
adventure does Odysseus need to deceive his wife? They are alone and nobody else is
around, so he could reveal his identity easily.

However,
there are three possible answers we can give. The first lies in the nature of the
character of Odysseus. Throughout the story, he is a character who is presented as being
in love with deception. This is something that even Athene notices and praises him for.
He is the kind of character that would rather deceive and show his cunning than tell the
truth. Note how Athene had to command him to reveal himself to his son, Telemachus. The
suggestion is that Odysseus would not have done so if it were left up to
him.

Secondly, I think there is a sense in which Odysseus
is very aware that he has been away for so long and his beautiful wife has been
surrounded by many suitors. He wants to ensure her faithfulness to him but also see how
she has changed.

Finally, Odysseus definitely wants to take
advantage of his disguise to know what situation he is going to have to face in the
inevitable showdown. He does not want to face the suitors without knowing more about
them, and perhaps Penelope might show a change in her personality if she knew that her
husband was actually there. Much better then to deceive her until a time when he is
ready to challenge the suitors.